The consistency of non-CH in the second-order constructible universe with Woodin cardinals

Let $C^2(\alpha9\omega)$ denote the inner model obtained from second-order definability over $\alpha9\omega$, let $\text{CH}$ denote the continuum hypothesis, and let $M_n$ denote the relevant inner model with $n$ Woodin cardinals. Consistency conjecture. For every $n$, $( eg\text{CH})^{C^2(\alpha9omega)}$ is consistent with $n$ Woodin cardinals, assuming the existence of $M_n$. This concerns the consistency strength of failures of the continuum hypothesis inside the second-order constructible universe; the supplied text gives no resolution status.